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Article

Oscillation of Bounded Solutions of Delay Differential Equations

Department of Mathematics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Letná 9, 042 00 Košice, Slovakia
Mathematics 2025, 13(1), 49; https://doi.org/10.3390/math13010049
Submission received: 21 November 2024 / Revised: 15 December 2024 / Accepted: 23 December 2024 / Published: 26 December 2024
(This article belongs to the Section C1: Difference and Differential Equations)

Abstract

:
The purpose of this paper is to give new oscillation criteria for second-order delay differential equations y ( t ) = p ( t ) y ( τ ( t ) ) . We introduce a new technique for the elimination of bounded nonoscillatory solutions.

1. Introduction

We consider the second-order functional differential equation with delayed argument:
y ( t ) = p ( t ) y ( τ ( t ) ) .
Throughout this paper, it is assumed that
  • (H1) p ( t ) C ( [ t 0 , ) ) , p ( t ) > 0 ,
  • (H2) τ ( t ) C 1 ( [ t 0 , ) ) , τ ( t ) < t , τ ( t ) > 0 , lim t τ ( t ) = .
As usual, by a proper solution of Equation (E), we mean a function y : [ t 0 , ) ( , ) which satisfies (E) for all sufficiently large t and sup { y ( t ) : t T } > 0 for all T t 0 .
Definition 1.
A proper solution is called oscillatory if it has arbitrary large zero; otherwise, it is called nonoscillatory.
There are two eventual cases for nonoscillatory solutions of (E), namely if y ( t ) is a nonoscillatory solution of (E), then there exists a number j { 0 , 2 } such that
y ( t ) y ( i ) ( t ) > 0 for 0 i j , ( 1 ) i y ( t ) y ( i ) ( t ) > 0 for j i 2 .
Definition 2.
Such y ( t ) is said to be a solution of degree j, and the totality of solutions of degree j is denoted by N j .
Therefore, if we denote the set of all nonoscillatory solutions of (E) by N , then N has the following decomposition:
N = N 0 N 2 .
The problem of determining oscillation criteria for particular functional differential equations has been a very active research area in the past few decades, and some of them are mentioned in references [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. The qualitative study of such equations has, besides its theoretical interest, significant practical importance. This is due to fact that differential equations arise in various phenomena, including in problems concerning electric networks containing lossless transmission lines (as seen in high-speed computers, where such lines are used to interconnect switching circuits), in the study of vibrating masses attached to an elastic bar, and in the solution of variational problems with time delays. We refer the reader to [22,23] for models from mathematical biology where oscillation and/or delay actions may be formulated by means of cross-diffusion terms.
Our aim is to present a new technique for the investigation of asymptotic properties of (E). For this, we recall the classical result of Koplatadze and Chanturia [16].
Lemma 1.
Assume that ( H 1 ) and ( H 2 ) holds true. If
lim sup t τ ( t ) t ( s τ ( t ) ) p ( s ) d s > 1 ,
then N 0 = for (E).
This result has been extended to more general differential equations (see, e.g., [9] or [19]). But we are about to make meaningful progress exactly for (E). Baculikova (see [3]) suggested two ways for improving Theorem 1. She noticed that the main idea of Koplatadze and Chanturia’s proof consists in the following estimates for possible solutions of degree 0:
y ( τ ( t ) ) = τ ( t ) ( s τ ( t ) ) p ( s ) y ( τ ( s ) ) d s   τ ( t ) t ( s τ ( t ) ) p ( s ) y ( τ ( s ) ) d s   y ( τ ( t ) ) τ ( t ) t ( s τ ( t ) ) p ( s ) d s .
Firstly, Koplatadze and Chanturia used the fact that y ( t ) is decreasing (see the third line of (3)), and Baculikova provided natural improvement in the sense of establishing a new monotonicity for y ( t ) in the form of α ˜ ( t ) y ( t ) , decreasing for a suitable function α ˜ ( t ) such that α ˜ ( t ) as t .
Secondly, the information about function ( s τ ( t ) ) p ( s ) y ( τ ( s ) ) is lost on the interval ( t , ) (see the second line of (3)), and Baculikova eliminates this insufficiency by establishing “the opposite” monotonicity of y ( t ) , in the sense that β ( t ) y ( t ) is increasing for certain function β ( t ) . In this paper, we introduce a new idea for preserving information about ( s τ ( t ) ) p ( s ) y ( τ ( s ) ) over the interval ( t , ) by establishing the important estimate
t ( s τ ( t ) ) p ( s ) y ( τ ( s ) ) d s τ ( t ) t ( s τ ( t ) ) p ( s ) y ( τ ( s ) ) d s
for a certain positive constant > 0 . Moreover, we improve Baculikova’s monotonicity.

2. Preliminary Results

In this section, we recall some results from Baculikova [3].
Lemma 2.
If
t 0 s p ( s ) d s = ,
then every solution y ( t ) of degree 0 of (E) satisfies
y ( t ) 0 a s t .
Let us denote that
P ˜ ( t ) = τ ( t ) τ ( t ) t p ( s ) d s , α 0 ˜ ( t ) = P ˜ ( t ) and α ˜ ( t ) = e α 0 ˜ ( t ) .
Lemma 3.
Let y ( t ) be a positive solution of degree 0 for Equation (E). Then,
α ˜ ( t ) y ( τ ( t ) ) i s a d e c r e a s i n g f u n c t i o n .
We introduce the auxiliary function ξ ( t ) C 1 ( [ t 0 , ) ) such that
ξ ( ξ ( t ) ) = τ 1 ( t ) .
Moreover, we employ the following supplementary functions:
P 1 ( t ) = α ˜ ( τ ( ξ ( t ) ) ) t ξ ( t ) s t α ˜ ( τ ( s ) ) p ( s ) d s , P 2 ( t ) = α ˜ ( t ) ξ ( t ) τ 1 ( t ) s t α ˜ ( τ ( s ) ) p ( s ) d s , P 3 ( t ) = α ˜ ( ξ ( t ) ) τ 1 ( t ) τ 1 ( ξ ( t ) ) s t α ˜ ( τ ( s ) ) p ( s ) d s .
It is assumed that there exist positive constants P i * , i = 1 , 3 such that
P 1 * ( t ) = P 1 ( t ) 1 P 2 ( t ) P 1 * , P 3 * ( t ) = P 3 ( t ) 1 P 2 ( t ) P 3 * .
Lemma 4.
Assume that there exists a function ξ ( t ) C 1 ( [ t 0 , ) ) satisfying (8) and that y ( t ) is a positive solution of degree 0 of (E); then,
y ( τ ( t ) ) K y ( t ) ,
where
K = 1 2 P 3 * P 1 * ( P 1 * ) 2 .
If it happens that
1 2 P 3 * P 1 * 0 ,
then the class N 0 = .

3. Results

In deriving the main results, we will rely on the Leibniz integral rule for differentiation under the integral sign, which claims that
d d t a ( t ) b ( t ) f ( s , t ) d s   = f ( b ( t ) , t ) d d t b ( t ) f ( a ( t ) , t ) d d t a ( t ) + a ( t ) b ( t ) t f ( s , t ) d s .
Lemma 5.
Assume that K > 0 is defined by (12) and
τ ( t ) p ( τ ( t ) ) p ( t ) A , A > 0 .
Then, there exists a constant > 0 such that, for any positive solution y ( t ) N 0 of (E),
t p ( s ) y ( τ ( s ) ) d s τ ( t ) t p ( s ) y ( τ ( s ) ) d s ,
where
(i)
if K A 1 , then ℓ is an optional positive number;
( i i )
if K A > 1 , then = 1 K A 1 .
Proof. 
Assume that y ( t ) N 0 is a positive solution of (E). Then,
y ( t ) 0 as t .
An integration of (E) yields
y ( τ ( t ) ) = τ ( t ) p ( s ) y ( τ ( s ) ) d s .
We consider the auxiliary function
h ( t ) = t p ( s ) y ( τ ( s ) ) d s τ ( t ) t p ( s ) y ( τ ( s ) ) d s .
It follows from (16) and (17) that h ( ) = 0 . Consequently, to verify (15), it is sufficient to show that h ( t ) 0 . The Leibniz integral rule implies that
h ( t ) = p ( t ) y ( τ ( t ) ) p ( t ) y ( τ ( t ) ) p ( τ ( t ) ) y ( τ ( τ ( t ) ) ) τ ( t )
which, in view of (11), provides
h ( t ) p ( t ) y ( τ ( t ) ) ( 1 + ) + K p ( τ ( t ) ) p ( t ) τ ( t ) .
Taking (14) into account, one obtains
h ( t ) p ( t ) y ( τ ( t ) ) ( 1 + ) + K A 0 .
The proof is complete now. □
Lemma 5 permits us to improve the monotonicity presented in (7).
We consider > 0 such as in Lemma 5. We denote that
P ( t ) = ( 1 + ) τ ( t ) α ˜ ( t ) τ ( t ) t p ( s ) α ˜ ( s ) d s , α 0 ( t ) = P ( t ) and α ( t ) = e α 0 ( t ) .
Lemma 6.
Let y ( t ) be a positive solution of degree 0 for Equation (E). Then,
α ( t ) y ( τ ( t ) ) i s a d e c r e a s i n g f u n c t i o n .
Proof. 
Assume that y ( t ) N 0 is a positive solution of (E). Setting (15) into (17), we have
y ( τ ( t ) ) ( 1 + ) τ ( t ) t p ( s ) y ( τ ( s ) ) d s   ( 1 + ) α ˜ ( t ) y ( τ ( t ) ) τ ( t ) t p ( s ) α ˜ ( s ) d s ,
where we have used (7). The last inequality is equivalent to
y ( τ ( t ) ) τ ( t ) + P ( t ) y ( τ ( t ) ) 0
which gives
[ y ( τ ( t ) ) α ( t ) ] 0 ,
and we can conclude that y ( τ ( t ) ) α ( t ) is a decreasing function. □
Now, we are prepared to establish new criteria for N 0 = for (E), or in other words, we are prepared for every bounded solution of (E) to be oscillatory.
Theorem 1.
Let (4) hold and the positive constants K, A, ℓ be the same as in Lemma 5. Assume that there exists a function ξ ( t ) C 1 ( [ t 0 , ) ) satisfying (8) and that α ( t ) is defined by (18). If
lim sup t α ( t ) τ ( t ) t p ( s ) ( s τ ( t ) ) α ( s ) d s > 1 1 + ,
then N 0 = for (E).
Proof. 
Assume, on the contrary, that (E) possesses a positive solution y ( t ) N 0 . Integrating (E) twice from t to and changing the order of integration, we obtain
y ( t ) t p ( s ) y ( τ ( s ) ) ( s t ) d s
Hence,
y ( τ ( t ) ) τ ( t ) t p ( s ) y ( τ ( s ) ) ( s τ ( t ) ) d s + t p ( s ) y ( τ ( s ) ) ( s τ ( t ) ) d s .
To simplify our notation, we employ the following functions:
F ( t ) = τ ( t ) t p ( s ) y ( τ ( s ) ) ( s τ ( t ) ) d s , G ( t ) = t p ( s ) y ( τ ( s ) ) ( s τ ( t ) ) d s ,
We claim that G ( t ) F ( t ) . To verify this, we employ a supplementary function:
r ( t ) = G ( t ) F ( t )
And we shall show that r ( t ) 0 . It follows from (5) and (22) that r ( ) = 0 , and therefore, it is sufficient to confirm that r ( t ) 0 . By the Leibniz integral rule,
r ( t ) = p ( t ) y ( τ ( t ) ) ( t τ ( t ) ) τ ( t ) t p ( s ) y ( τ ( s ) ) d s   p ( t ) y ( τ ( t ) ) ( t τ ( t ) ) + τ ( t ) τ ( t ) t p ( s ) y ( τ ( s ) ) d s   = ( + 1 ) p ( t ) y ( τ ( t ) ) ( t τ ( t ) )   + τ ( t ) τ ( t ) t p ( s ) y ( τ ( s ) ) d s t p ( s ) y ( τ ( s ) ) d s 0 ,
where we have used (15). Finally, (22), together with (24), implies that
y ( τ ( t ) ) ( 1 + ) τ ( t ) t p ( s ) y ( τ ( s ) ) ( s τ ( t ) ) d s   ( 1 + ) ) α ( t ) y ( τ ( t ) ) τ ( t ) t p ( s ) ( s τ ( t ) ) α ( s ) d s .
where (19) has been also employed. Since (26) contradicts (21), the proof is finished. □
This is the standard way to illustrate the progress achieved by means of Euler differential equations.
Example 1.
Consider the differential equation
y ( t ) = p 0 t 2 y ( λ t ) , p 0 > 0 , λ ( 0 , 1 ) .
By Theorem 1, Equation (E1) for λ = 0.5 has no noscillatory solutions from class N 0 , provided that p 0 > 5.177 . On the other hand, it is easy to verify that
P ˜ ( t ) = p 0 t ( 1 λ ) a n d α 0 ˜ ( t ) = p 0 ( 1 λ ) ln t
Hencez,
α ˜ ( t ) = t β 0 , w h e r e β 0 = p 0 ( 1 λ ) .
We set ξ ( t ) = λ t . Consequently,
P 1 ( t ) = p 0 λ β 0 2 1 λ β 0 2 β 0 + λ 1 + β 0 2 1 1 + β 0 ,
P 2 ( t ) = p 0 λ β 0 2 1 λ β 0 2 β 0 + λ 2 + β 0 2 λ 1 2 1 + β 0 ,
P 3 ( t ) = p 0 λ β 0 2 1 λ β 0 2 β 0 + λ 3 + β 0 2 λ 1 + β 0 .
Then,
= λ K λ
and
α ( t ) = t γ , w h e r e γ = p 0 ( 1 + ) λ β 0 λ 1 + β 0 .
Condition (21) reduces for (E1) to
p 0 λ γ 1 γ + λ λ γ 1 + γ > 1 + .
Consequently, for λ = 0.5 , condition (27) is satisfied for p 0 = 2.496 , which by Theorem 1 yields N 0 = for (E1). On the other hand, Baculikova’s way [3] requires p 0 = 2.56 . The progress is obvious.

4. Discussion

In this paper we have introduced a new method for investigating the properties of linear second-order delay differential equations. There may be two ways for generalizing or improving the presented result. The first way would be to try one’s hand at establishing the iteration process for improving the monotonicity (19) (see the technique presented in [12]). The second way would involve trying to extend the results to higher-order differential equations.

Funding

This research received no external funding.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

The author declares no conflicts of interest.

References

  1. Agarwal, R.P.; Grace, S.R.; O’Regan, D. Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations; Kluver Academic Publishers: Dordrecht, The Netherlands, 2002. [Google Scholar]
  2. Agarwal, R.P.; Zhang, C.H.; Li, T. Some remarks on oscillation of second order neutral differential equations. Appl. Math. Comput. 2016, 274, 178–181. [Google Scholar] [CrossRef]
  3. Baculikova, B. Monotonic properties of Kneser solutions of second order linear differential equations with delayed argument. Opusc. Math. 2025, 45, 27–38. [Google Scholar] [CrossRef]
  4. Baculikova, B. Oscillation of second-order nonlinear noncanonical differential equations with deviating argument. Appl. Math. Lett. 2019, 91, 68–75. [Google Scholar] [CrossRef]
  5. Baculikova, B. Oscillatory behavior of the second order noncanonical differential equation. Electron. J. Qual. Theory Differ. Equ. 2019, 89, 1–17. [Google Scholar] [CrossRef]
  6. Baculikova, B. Oscillation and Asymptotic Properties of Second Order Half-Linear Differential Equations with Mixed Deviating Arguments. Mathematics 2021, 9, 2552. [Google Scholar] [CrossRef]
  7. Li, T.X.; Baculikova, B.; Dzurina, J. Oscillation results for second-order neutral differential equations of mixed type. Tatra Mt. Math. Publ. 2010, 48, 101. [Google Scholar] [CrossRef]
  8. Bohner, M.; Grace, S.R.; Jadlovská, I. Oscillation criteria for second-order neutral delay differential equations. Electron. J. Qual. Theory Differ. Equ. 2017, 60, 1–12. [Google Scholar] [CrossRef]
  9. Dzurina, J. Oscillation of second-order trinomial differential equations with retarded and advanced arguments. Appl. Math. Lett. 2024, 153, 1–8. [Google Scholar] [CrossRef]
  10. Tamilvanan, S.; Thandapani, E.; Dzurina, J. Oscillation of second order nonlinear differential equation with sb-linear neutral term. Differ. Equ. Appl. 2017, 9, 1–7. [Google Scholar]
  11. Chatzarakis, G.; Jadlovská, I. Improved oscillation results for second-order half-linear delay differential equations. Hacet. J. Math. Stat. 2019, 48, 170–179. [Google Scholar] [CrossRef]
  12. Jadlovska, I.; Dzurina, J. Kneser-type oscillation criteria for second-order half-linear delay differential equations. Appl. Math. Comput. 2020, 380, 1–15. [Google Scholar] [CrossRef]
  13. Jadlovska, I. Oscillation criteria of Kneser-type for second-order half-linear advanced differential equations. Appl. Math. Lett. 2020, 106, 1–8. [Google Scholar] [CrossRef]
  14. Kiguradze, I.T.; Chanturia, T.A. Asymptotic Properties of Solutions of Nonatunomous Ordinary Differential Equations; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1993. [Google Scholar]
  15. Koplatadze, R.; Kvinkadze, G.; Stavroulakis, I.P. Properties A and B of n-th order linear differential equations with deviating argument. Gorgian Math. J. 1999, 6, 553–566. [Google Scholar] [CrossRef]
  16. Koplatadze, R.; Chanturia, T.A. On Oscillatory Properties of Differential Equations with Deviating Arguments; Tbilisi University Press: Tbilisi, Georgia, 1977. [Google Scholar]
  17. Kusano, T. On even order functional differential equations with advanced and retarded arguments. Differ. Equ. 1982, 45, 75–84. [Google Scholar] [CrossRef]
  18. Kusano, T. Oscillation of even order linear functional differential equations with deviating arguments of mixed type. J. Math. Anal. Appl. 1984, 98, 341–347. [Google Scholar] [CrossRef]
  19. Kusano, T.; Lalli, B.S. On oscillation of half-linear functional differential equations with deviating arguments. Hiroshima Math. J. 1994, 24, 549–563. [Google Scholar] [CrossRef]
  20. Laddas, G.; Lakshmikantham, V.; Papadakis, J.S.; Zhang, B.G. Oscillation of higher-order retarded differential equations generated by retarded argument. In Delay and Functional Differential Equations and Their Applications; Academic Press: New York, NY, USA, 1972; pp. 219–231. [Google Scholar]
  21. Li, T.; Rogovchenko, Y. Oscillation of second-order neutral differential equations. Math. Nachr. 2015, 288, 1150–1162. [Google Scholar] [CrossRef]
  22. Li, T.; Pintus, N.; Viglialoro, G. Properties of solutions to porous medium problems with different sources and boundary. Z. Angew. Math. Phys. 2019, 70, 86. [Google Scholar] [CrossRef]
  23. Li, T.; Viglialoro, G. Boundedness for a nonlocal reaction chemotaxis model even in the attreaction-dominated regime. Differ. Equ. 2021, 34, 315–336. [Google Scholar]
  24. Naito, M. Oscillation Criteria for Second Order Ordinary Differential Equations. Canad. Math. Bull. 2020, 63, 276–286. [Google Scholar] [CrossRef]
  25. Wu, Y.; Yu, Y.; Xiao, J. Oscillation of Second Order Nonlinear Neutral Differential Equations. Mathematics 2022, 10, 2739. [Google Scholar] [CrossRef]
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Dzurina, J. Oscillation of Bounded Solutions of Delay Differential Equations. Mathematics 2025, 13, 49. https://doi.org/10.3390/math13010049

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Dzurina J. Oscillation of Bounded Solutions of Delay Differential Equations. Mathematics. 2025; 13(1):49. https://doi.org/10.3390/math13010049

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Dzurina, Jozef. 2025. "Oscillation of Bounded Solutions of Delay Differential Equations" Mathematics 13, no. 1: 49. https://doi.org/10.3390/math13010049

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Dzurina, J. (2025). Oscillation of Bounded Solutions of Delay Differential Equations. Mathematics, 13(1), 49. https://doi.org/10.3390/math13010049

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